# Double Pendulum: Chaos in the Physics Lab!

With a few pieces of wood and the resolve to go beyond theory, Ramkrishna Joshi, Anoosha C, and Durga Machavallavan built a double pendulum. And their project just won them national recognition.

It all began with hacksaw blades. Traditionally used to cut metallic and non-metallic objects quickly and precisely, these tools also became popular among do-it-yourself experimentalists because of their inexpensive nature and versatility.

That’s probably why they were present in the laboratories of Azim Premji University where Ramkrishna Joshi used to spend most of his time, while doing his undergraduate studies in physics. While tinkering with a hacksaw blade one day, Ramkrishna realised that he could use them to recreate an experiment that had caught his fancy back when he was in Grade XII, the double pendulum.

A pendulum is familiar to most of us; picture a swing, a grandfather clock, or simply a bob on a string. It is one of the easiest ways to demonstrate the concept of simple harmonic motion and to measure gravity. Once you let go of the bob at the end of a string, it is easy to visualise the path it is going to take. It will swing left to right at a more-or-less constant speed until gravity causes it to slow down and eventually stop.

Like most of us, Ramkrishna too learnt about the simple pendulum in high school. However, while in Grade XII, he was provoked by a book on non-linear dynamics by American mathematician Steven Strogatz to wonder what happens in the case of two back-to-back pendulums.

Scientists and mathematicians have been mesmerised by the idea of a double pendulum since the 16th century at least, and it is easy to see why. It turns out that affixing a second pendulum at the end of the first changes things dramatically. The system is no longer following a repetitive to-and-fro path; in fact, it is accurate to say that it goes completely bonkers!

#### Deceptively predictable

The bizarre behaviour of a double pendulum intrigued Ramkrishna. It just seemed like a cool experiment and I wanted to make one myself. Each pendulum affects the other, and you end up with these wonderfully complicated trajectories,” he says. When he realised one day in the laboratory that hacksaw blades were capable of oscillating like a pendulum, Ramkrishna knew this was his chance.

To the untrained eye, the erratic movement of a double pendulum would come off as completely random. Each time the system is set into motion, it seems to take a completely different path — even if the conditions, such as the height at which it was released, are more or less the same.

What exactly was happening to the laws of physics that made the double pendulum behave this way? It was Swiss physicist Daniel Bernoulli who recorded the first analysis on this, way back in 1733. Ultimately, it would be French mathematician Jean le Rond d’Alembert who successfully came up with the differential equations that would describe the movement of a double pendulum.

The fact that there are equations that describe its movement indicates that the double pendulum’s motion is not random; meaning that as haphazardly as it may seem to move, it can be predicted. So then why is it so hard to get a double pendulum to move the same way twice? It all comes down to chaos.

To illustrate how a chaotic system differs from a traditional one, Ramkrishna brings up a scenario of an athlete running at a specific speed on a track. If the athlete starts at a fixed point A, she will always reach a fixed point B after a specific amount of time. A small deviation from her starting point would only mean a similarly small deviation from point B. However, if this was a chaotic system, even a very small deviation from A would mean that she ends up following a completely different path and destination point.

Today, chaos theory has applications in everything from population biology, cryptography, psychology, economics and polymer chemistry.

Another example of chaotic behaviour is weather. There are so many intermingled factors affecting weather patterns that meteorologist Edward Lorenz famously suggested that the flap of a butterfly’s wings in Brazil could set off a tornado in Texas, an insight that would spark off the field of chaos theory. Today, chaos theory has applications in everything from population biology, cryptography, psychology, economics and polymer chemistry.

#### Designing a chaotic system

The double pendulum is one of the simplest and neatest examples of chaos. These systems are extremely sensitive to initial conditions — so much that even if things change by a bit, you’re going to get a completely different outcome,” says Ramkrishna. Reading about all this was fascinating enough, but he wanted to go one step further and see the chaos for himself.

So would the hacksaw blades work?

Teaming up with his classmates Anoosha C and Durga M, he started by attaching two of the blades. But it quickly became apparent that hacksaw blades were not working. They then tried the same with thick acrylic perspex sheets, but like the blades, they were too flexible. Clearly, many more factors needed to be considered. For a successful model, the pendulum material, its length and mass have to be such that they cannot crash against each other, and have to oscillate long enough to minimise friction and other dampening effects. They would need a more rigid material.

The physics honours lab at the University was at the time still in the making. Looking back, Ramkrishna realises this was a good thing. Things were scattered around, and we had the freedom to wander around, find objects here and there, and build something. That was how we were able to build the final version of the double pendulum,” he says.

It took them over three months to finalise the materials they would use. It was ultimately wood that did the trick, hinged together with the help of special friction-minimising ball bearings procured for them by their professor, Anish Mokashi. Figuring out optimal parameters of the wooden limbs, and fabricating the set-up was not easy, but they managed it with the help of a lab associate, Md Hossen Mondal.

#### Simulating, presenting, winning

When the students began their Waves’ course in physics, an opportunity came to present what they had been working on. Meanwhile, the team approached another professor, Kaustubh Manchanda, to help them write a code so that they could develop a computer simulation version of their experiment, in which the conditions were under their control. It helped tremendously that Durga had expertise in simulation techniques.

Why the need for simulations? Ramkrishna says, The beauty of the computer is that we can tell it exactly what the length of each pendulum should be and their masses, and it will show the evolution of the system. The uncertainty of real life is that you can never match this condition. I might set the length in the computer to 5 cm but in real life, it may be 5.001 cm. A chaotic system is such that even this difference is enough for the two paths to completely deviate after a few seconds. A comparative study helped us witness chaos in real life.”

The trio’s adventure with the double pendulum culminated in Jan. 2024, when they signed up to compete in IIT Bombay’s student research festival ResCon 2024. Their abstract was selected and Ramkrishna headed to Mumbai to represent the team. After a gruelling two days of presentation and facing the scrutiny of senior scientists, he was delighted when their project was adjudged the best one in the theoretical physics and computation category. It also made it to the overall list of 10 best projects across categories. Ramkrishna was elated to accept the reward from nuclear physicist and former chairman of the Atomic Energy Commission of India, Anil Kakodkar.

When I was at Azim Premji University, I used to be in the lab pretty much the whole day. I was not doing this for a project or anything, but just to explore. Hundreds of books tell us the equations for something, we know exactly what will happen — theoretically. But we need to verify this because we cannot just take for granted that a theory is the truth.”

Ramkrishna Joshi, BSc Physics (2020-2023)

Though tempted to take their experimental set-up with them, the students decided to leave it behind in the laboratory for the benefit of their juniors who may want to build upon their work. Also, sometimes school students visit the campus,” Ramkrishna adds. We hope that they see our double pendulum, play with it, and try to understand it better.”

The original motivation behind doing this experiment for Ramkrishna, who is now pursuing a Master’s at IIT Hyderabad, was just to have fun. When I was at Azim Premji University, I used to be in the lab pretty much the whole day. I was not doing this for a project or anything, but just to explore. Hundreds of books tell us the equations for something, we know exactly what will happen — theoretically. But we need to verify this because we cannot just take for granted that a theory is the truth.”